Your data matches 26 different statistics following compositions of up to 3 maps.
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St000290: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 2
011 => 0
100 => 1
101 => 1
110 => 2
111 => 0
0000 => 0
0001 => 0
0010 => 3
0011 => 0
0100 => 2
0101 => 2
0110 => 3
0111 => 0
1000 => 1
1001 => 1
1010 => 4
1011 => 1
1100 => 2
1101 => 2
1110 => 3
1111 => 0
00000 => 0
00001 => 0
00010 => 4
00011 => 0
00100 => 3
00101 => 3
00110 => 4
00111 => 0
01000 => 2
01001 => 2
01010 => 6
01011 => 2
01100 => 3
01101 => 3
01110 => 4
01111 => 0
10000 => 1
10001 => 1
10010 => 5
10011 => 1
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
St000293: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 1
011 => 0
100 => 2
101 => 1
110 => 2
111 => 0
0000 => 0
0001 => 0
0010 => 1
0011 => 0
0100 => 2
0101 => 1
0110 => 2
0111 => 0
1000 => 3
1001 => 2
1010 => 3
1011 => 1
1100 => 4
1101 => 2
1110 => 3
1111 => 0
00000 => 0
00001 => 0
00010 => 1
00011 => 0
00100 => 2
00101 => 1
00110 => 2
00111 => 0
01000 => 3
01001 => 2
01010 => 3
01011 => 1
01100 => 4
01101 => 2
01110 => 3
01111 => 0
10000 => 4
10001 => 3
10010 => 4
10011 => 2
Description
The number of inversions of a binary word.
Matching statistic: St000228
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> []
=> 0
1 => [1,1] => [[1,1],[]]
=> []
=> 0
00 => [3] => [[3],[]]
=> []
=> 0
01 => [2,1] => [[2,2],[1]]
=> [1]
=> 1
10 => [1,2] => [[2,1],[]]
=> []
=> 0
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> 0
000 => [4] => [[4],[]]
=> []
=> 0
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
010 => [2,2] => [[3,2],[1]]
=> [1]
=> 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
100 => [1,3] => [[3,1],[]]
=> []
=> 0
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 0
0000 => [5] => [[5],[]]
=> []
=> 0
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 3
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 4
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
1000 => [1,4] => [[4,1],[]]
=> []
=> 0
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 0
00000 => [6] => [[6],[]]
=> []
=> 0
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 4
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 3
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 6
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 5
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 4
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 6
01000 => [2,4] => [[5,2],[1]]
=> [1]
=> 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 4
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 3
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 5
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 4
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
10000 => [1,5] => [[5,1],[]]
=> []
=> 0
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 4
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000355: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 0
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,2,5,4,6] => 4
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 3
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => 3
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 4
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => 5
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 6
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 4
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 5
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 4
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => 2
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000369
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
St000369: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 3
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 6
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 5
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 6
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 4
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 5
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 4
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 4
Description
The dinv deficit of a Dyck path. For a Dyck path $D$ of semilength $n$, this is defined as $$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$ In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$. See also [[St000376]] for the bounce deficit.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000462: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 0
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,2,5,4,6] => 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => 1
Description
The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000589
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000589: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 4
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 4
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 4
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 2
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000597
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000597: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block.
Matching statistic: St000605
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000605: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000609
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000609: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 4
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 4
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 4
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 2
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000747A variant of the major index of a set partition. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001175The size of a partition minus the hook length of the base cell. St001377The major index minus the number of inversions of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001438The number of missing boxes of a skew partition. St000516The number of stretching pairs of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000567The sum of the products of all pairs of parts. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000264The girth of a graph, which is not a tree.